Abstracts

Invited Talks

Keiran Fleming - A cellular diagonal approximation for the associahedra

The associahedra, or Stasheff polytopes, are a central object in homotopical algebra as the cellular chain complexes on the associahedra define the d.g. A-infinity operad and thus model homotopy associative structures. In this talk we define an operad structure on particular realisations of the associahedra and construct a compatible diagonal approximation. If time permits some potential applications of the diagonal formula will be considered.

Ana Garcia-Pulido - Twisted spin and almost even-Clifford structures

Twisted spin structures and almost even Clifford structures have recently opened a path to new algebraic structures on Riemannian manifolds. The first are a generalisation of Spin, Spin^c, Spin^q structures whereas the second generalise almost Hermitian and quaternionic Hermitian structures.

In this talk, I will first explain the notion of a twisted Spin structure and summarise some geometric facts analogous to those existing in classical geometry. Motivated by this definition, I will describe the concept of an almost even-Clifford Hermitian structure and the relationship with (twisted) Spin structures. This work is joint with Arizmendi and Herrera.

Jelena Grbic - Homology theory of super-hypergraphs

Hypergraphs can be seen as incomplete abstract simplicial complexes in the sense that taking subsets is not a closed operation in hypergraphs. This notion can be extended to ∆-sets with face operations only partially defined, these objects we name super-hypergraphs. In this talk I will set foundations of homology theory of these combinatorial objects.

Lennart Meier - Chromatic Localizations of Algebraic K-Theory

Algebraic K-theory is a fundamental, but difficult invariant of rings and ring spectra. A by now classic approach to understand difficult spaces or spectra is to apply localization functors to them. We will focus on localizations with respect to Morava K-theories, resulting in the chromatic tower and allowing us to define the height of a spectrum. Rognes has observed that the height of a ring spectrum appears to shift by one by applying K-theory, a phenomenon called red shift. Presenting joint work with Land and Tamme, I will speak about some recent new results in this direction.

Luca Pol - Global representation theory of finite groups

Fix k a field of characteristic and let G denote the category of finite groups and conjugacy classes of epimorphisms. The goal of this talk is to describe the complexity of the abelian category of contravariant functors from G to the category of -vector spaces. Roughly speaking, we consider sequences of k-vector spaces, indexed by all finite groups, each of which is equipped with an action of the outer automorphism group. This category generalizes the category of VI-modules appearing in stable representation theory, as introduced by Church and Farb, and connects to global equivariant stable homotopy theory, as introduced by Schwede. As an application of our work, we completely describe the Balmer spectrum of the rational global stable homotopy category for the global family of elementary abelian p-groups. This is joint work with Neil Strickland.

Constanze Roitzheim - Equivariant homotopy commutativity, trees and chicken feet

Commutativity up to homotopy can be daunting, and it gets even more difficult to track when equivariant structures get introduced. In the case of a finite group, however, the options for equivariant homotopy commutativity can be encoded using simple combinatorics, and we will show some examples.

Bernadette Stolz - Outlier-robust subsampling techniques for persistent homology

The amount and complexity of biological data has increased rapidly in recent years with the availability of improved biological tools. Topological data analysis and more specifically persistent homology have been successfully applied to biological settings. When attempting to study large data sets however, many of the currently available algorithms fail due to computational complexity preventing many interesting biological applications. De Silva and Carlsson (2004) introduced the so called Witness Complex that reduces computational complexity by building simplicial complexes on a small subset of landmark points selected from the original data set. The landmark points are chosen from the data either at random or using the so called maxmin algorithm. These approaches are not ideal as the random selection tends to favour dense areas of the point cloud while the maxmin algorithm often selects outliers as landmarks. Both of these problems need to be addressed in order to make the method more applicable to data. Chawla (2013) developed a version of k- means that detects outliers while clustering data points. We show how this method can be used to select landmarks for persistent homology and also propose another new method specifically for the use in topological data analysis that can detect outliers based on the local persistent homology around data points. We show how both of these methods outperform the existing subsampling methods for persistent homology. We further illustrate how local persistent homology can be applied to detect geometric anomalies in data.

Jamie Walton - Topological invariants for aperiodic patterns

A periodic pattern of Euclidean space, such as an infinite chequerboard decoration in the plane, is essentially determined by its space group of global symmetries. Quite recently there has been a surge of interest in aperiodically ordered patterns, which never precisely repeat themselves but frequently ‘almost repeat’. Such objects are used as models for quasicrystals, and arise as natural objects associated to certain dynamical and number theoretic constructions. Being notably short of global symmetries, these objects require new mathematical tools to study them. The approximate symmetry of these patterns is effectively captured by the topology of associated moduli spaces of patterns. In this talk I will introduce the field of Aperiodic Order and how one may use Algebraic Topology to define natural invariants for aperiodic patterns. I will discuss recent joint work with John Hunton, which introduces new techniques for analysing both their translational and rotational structure. Applied in the periodic setting, one may recover the classical space group of global symmetries. For special ‘model sets’, used as mathematically idealised quasicrystals, we obtain an invariant which has a natural quotient isomorphic to the crystallographers’ aperiodic space group.

Ittay Weiss - Quantales in the foundations of topology and data analysis

If one is willing to slightly relax the meaning of what a metric is, primarily allowing the codomain to be a quantale, then all topological spaces are metrizable. This observation (Flagg, 1997) raises some questions. Is this metric formalism for topology just a curiosity? Since a metric space valued in a quantale Q is the same thing as a Q-enriched category where is the line between algebra and geometry? Are there quantales other than the usual one of the non-negative real numbers that may be of use in data analysis? In this talk we will address these questions, attempting to argue that this quantale metric approach offers a unifying formalism that blurs the distinction between qualitative and quantitative topology. Certain topological invariants will be shown to be the limiting case of metric invariants, criteria for the geometricity of a Q-valued metric space will be given, and, as time permits, applications to topological data analysis will be indicated.

Contributed Talks

Ilia Pirashvili - The fundamental groupoid of Stanley-Reisner Rings and binoid varieties

Mariam Pirashvili - Topology and geometry of molecular conformational spaces and energy landscapes.

Matthew Burfitt - Multi-parameter problems in topological data analysis

Jordan Williamson - Algebraic Models for Change of Groups Functors in Rational Equivariant Stable Homotopy Theory

Ulrich Pennig - Equivariant higher twisted K-theory of SU(n) via exponential functors